Theory of Heat to the Steam-engine. 251 



and we have 



^— u • • • '^^^ :wni^%^f.U ,•! ,«JW 



By this equation m is expressed as a function of T and v, 

 because u and cr are functions of T. 



12. In order to be able to apply equations (III) and (IV) to 



our case, we must next determine the magnitudes -p and -j^. 



If the volume of the vessel increases by dv, then the quantity 

 of heat which must be imparted to the mass in order to maintain 

 a constant temperature will be generally expressed by 



-j-dv. 

 dv 



But this quantity of heat is expended solely in the vaporization 

 which takes place during the expansion ; so that if r represents 

 the heat required to vaporize the unit of mass, the above quan- 

 tity of heat may also be represented by 



and we have 



But according to (7), 



hence 



dm , 



r—f-dVy 



dv 



dQ _ dm 

 dv ^ dv' 



dm 

 dv 



^ = ^ (8) 



dv u ^ ' 



Let u,s next assume, that whilst the volume of the vessel 

 remains constant, the temperature of the mass increases by c?T j 

 then the general expression for the requisite quantity of heat 

 will be 



dT"^^' 

 This quantity of heat consists of three parts : — 



(1) The liquid part M—m of the whole mass suffers an incre- 

 ment of temperature dT, for which, c being the specific heat of 

 the liquid, the quantity of heat 



{M-m)cdT 

 is necessary. . s 



(2) The vaporous part m will also undergo an increment of 

 temperature dT, but it will be thereby compressed so as still to 



